Causal relationship implies that their recurrence lead to the same consequences. However, a causal relationship can be subject to fluctuations due to random deviations. These fluctuations are due to the uncontrolled and unaccounted factors and are identified by statistical laws.

One of the most relevant issues of modern chemistry of biologically active substances is the problem of creating new effective radioprotectors. The main demands on these drugs are low effective dose, low toxicity and lack of side effects. The existence of side effects significantly limits the practical applicability of radioprotectors. Statistical methods are the most rational in solving problems that are associated with the study of action of a combination of factors on the biosystem. Since the effect of the interaction of drugs with biosystem depends on many conditions, then it has a probabilistic nature. Therefore it is preferable to use a probabilistic model.

It is not always possible to construct an adequate model which describes the relationship of the chemical structure of the compound with its biological activity. If the model is overloaded with a large number of non-essential characteristics use such model becomes almost impossible.

From the inequality (5) it follows that the chemical compound number N = 100 ( = 4.462) is not compatible with other the sample units. Consequently, the chemical compound is to be excluded from the sample and calculating the average value must be repeated. After recurrence the calculations, we have found that the chemical compounds numbered 96, 97, 98, and 99 also must be excluded from the sample. Now the average value has the following statistics:

N = 95,   = 2.80 ± 0.05,   = 2.235,   = 3.474,   S_z = 0.27,

                                                                                           (6)        

Here  and  are the minimum and maximum values of Z in sample that contains N = 95 units. Sample satisfies the following inequality:

 = 3.094 <  = 14.1,   p = 0.88,   N = 95.     

                                                                                             (7)

Thus, the sample is uniform and fits the normal distribution. Here p value determines the significance level of criterion which determines the probability of error (~ 10%); f is the number of degrees of freedom. Criterion of Wilk-Shapiro is also satisfied: W = 0.989 >   = 0.950.

Now we can determine the average value of Z for the effective and ineffective radioprotectors (N = 95). As a result, we obtained the following statistics:

N₁ = 57,  = 2.71 ± 0.06,   = 2.235,  = 3.300,  S_z1 = 0.24,

 N₂ = 38,   = 2.95 ± 0.08,  = 2.316,   = 3.474,  S_z2 = 0.27.                                                               (8)

Values of Z are located around  and  for the effective and ineffective chemical compounds, respectively. Using tabulated values of t - distribution, we can verify whether the distinction in the average values of Z sign ( > ) statistically significant. First, we compare the variances of samples:  = 1.34 < . That is, the distinction of the dispersions is not statistically significant. Then we use the following inequality [7]:

                                                         (9)

The inequality (9) shows that at the 5% significant level the null hypothesis of equality of average values can be rejected. Consequently, the difference between the average values     and are statistically significant.

In the first approximation, we can assume that the average value = 2.80 is a threshold that separates chemical compounds with different radioprotective efficiency. However, it is better to choose the threshold value through repeated testing various Z values close to  (for example, within the mean error). You can then use the value of Z which results to a more convincing statistical inference. This approach is demonstrated in the search of the classification rules by statistical methods for rates and proportions.

 According to the analysis, it is preferable to choose a threshold is equal to = 2.87.  Importantly, the chemical compounds (NN = 97 - 100) have the sign of Z noticeably larger than the average value  and therefore does not violate the inequality:  > .

We need to verify to see whether the separation of chemical compounds into two conditional groups is the result of random factors. Description of classifications, it is convenient to start with the construction of the table of mutual contingency (or association) [8,9] (cross-selection method). Figure 1 shows the distribution of the chemical compounds by quadrants of the rectangular 2 × 2 table (table of "four fields"). In each cell of the table is indicated the number (frequency) of q_ij objects. Obviously, the classification model better describes the phenomenon, the closer the contingency table to diagonal form. In which connection for the objects in each quadrant, we do not assume the existence of a functional mathematical relationship between the dependent variable and the explanatory variable.

Contingency (association) method is applicable, if the sample size satisfies the following inequality: . It is generally believed that the frequencies q_ij meet the inequality of  subject to i ≠ j  [8].

We use the following equation to determine the Pearson contingency coefficient Φ [9] between the radioprotective efficacy and value of the sign of Z  :

  .         (10)

Here number of degrees of freedom is equal to f = N – 2;  = 45 is number of effective chemical compounds having the sign value  subject to D 1 mM/kg;  = 12 is number of effective chemical compounds having the sign value  subject to D 1 mM/kg;  = 29 is number of effective chemical compounds having the sign value  subject to D > 2mM/kg;  = 14 is number of effective chemical compounds having the sign value  subject to D > 2mM/kg; (Table 2). We can also be used the Yule coefficient association for tetrachoric contingency tables [8]:

.                                  (11)

The coefficient Q = 0.77 point to the existence of the interrelation between the signs. Obviously, this coefficient is in the following range of values: .

Signs RE (the radioprotective efficiency) and Z are independent if the product of the marginal or unconditional proportions is equal to the joint proportion (see Table 2). For example, we obtained the following result: . These proportions differ considerably. The greater the distinction, the interdependence of signs RE and Z is greater.

The application of the threshold value leads to more convincing statistical results than using the average value of In brackets (see Table 2), we reported the statistical results that have been obtained for the average value is equal to . Using the average value also suggests a correlation signs at significance level α = 0.05. In this case, the strength of the interrelation too weak:  = 0.19. Therefore, it is preferable to use the threshold value 2.87. The adequacy of the model, we can verify using the value of the empirical error. The error is determined by the fraction of misclassified objects: . Using the data in Table 2, we found the following value of the empirical error of the model: . Application of the threshold value  reduces the empirical error of model (approximately 21%).

Let's see the representativeness of the sample (Table 1). Using a table of random numbers [6], we will make a partial sample of data Table 1. The method of random numbers avoids involuntary and systematic mistakes in the preparation of the sample. As a result, we obtained the following sequence of random numbers:

03, 47, 43, 73, 86, 36, 96, 46, 63, 71, 62, 33, 26, 16, 80, 45, 60, 11, 14, 10,  74, 24, 67, 42, 81, 57, 20, 53, 32, 37, 27, 07, 51, 79, 89, 76, 66, 56, 50, 90.                                           (12)

A series of random numbers, we can obtain, starting from any point of the table of random numbers. We wrote all the random numbers that do not exceed number of 96 [6]. Comparing these numbers with random numbers of chemical compounds Table 1, the partial sample of 40 items was obtained. In a partial sample the sequence of chemical compounds represented by "with an open mind" [10].  Statistics of the partial sample is as follows:

 N  = 40,     = 2.82 ± 0.07,    = 2.316,     = 3.300,    S_z = 0.23.

 N₁ = 24,     = 2.78 ± 0.08,     = 2.333,     = 3.300,    S_z1 = 0.21,

N₂ = 16,     = 2.88 ± 0.13,      = 2.316,     = 3.263,    S_z2 = 0.25.                                                         (13)

This result is similar to the statistics (6), at while the sign of Z is represented in the same proportion as in the original sample.

The standard error of contingency coefficient we can be assessed using the following equation:

                                                                                       (14)                                         

Testing of the significance is carried out by using chi - test [9]:  > = 3.84,                                                    (15)

i.e., at the α = 0.05 significance level of the null hypothesis can be rejected. For normally distributed data, you can additionally use the tetrachoric coefficient (-1 ≤  r_tet  ≤ 1) association:

 .                                                                                         (16)

However, if the distribution of frequencies on borders of two-by-two table is non-uniformly, then coefficient becomes unreliable. Therefore, commonly used [8,9], Pearson goodness of fit (adjusted for continuity of Yates):

= 19.9(10.8)  >  = 3.84.                                  (17)

Here N = q₁₁ + q₁₂ + q₂₂ + q₂₁ is the sum of all frequencies.  The inequality (17) shows that there is a statistically significant interrelation of signs. However, the criterion (17) does not give an idea of the strength of the signs interrelation. The assessment of closeness of the linkage between the signs can be obtained by using the coefficient of mutual contingency Pearson:

      .                                                              (18)

The indicator of mean-square of mutual conjugation  is equal to:

                                                                                           (19)

Using equation (18) we determine the coefficient of mutual contingency K = 0.43 (0.32), which confirms the interrelation of dichotomous signs.

Study of the interrelationship of the molecules structure - activity showed that the electronic sign of Z is associated with the Shannon informational function [11]:

,                                                       (20)

where , and the following ratios are met for : , ;  ,  is number of varieties of atoms in the molecule, N is the total number of atoms. The ratio  determines the relative share of i-th kind of the atom in the molecule [12]. Shannon function is an integral characteristic of the molecule that determines the measure of uncertainty (or diversity) of the structure of chemical compound. The smaller value of the function H, the more diverse (on the relative content of atoms) a multicomponent system.

Using the data of Table 1 we define the average value of the information function:

N  = 100,    = 1.80 ± 0.04,      = 1.174,      = 2.249,     S_H = 0.21.                                           (21)

We verify the compatibility of the units of the sample on the basis of H:

                                                                              (23)

Consequently, the sample does not contain incompatible units. Statistics of average values of information functions for effective radioprotectors will be as follows:

N₁ = 57,    = 1.76 ± 0.06,     = 1.378,     = 2.131,    S_H1 = 0.21.                                          (24)

This subset is close to a normal distribution: , and the following inequality  satisfies to the criterion of  Wilk-Shapiro:   W = 0.951 >   = 0.947. Let's see the compatibility of the units of this subset:

                                                                                           (25)

These inequalities are point to the lack of incompatible units.

For the inefficient radioprotectors statistics of the average value will be as follows:

N₂ = 43,     = 1.85 ± 0.06,    = 1.174,    = 2.249,   S_H2 = 0.20.                                                       (26)

Checking of units of the second subset leads to inequalities:

                                                                                          (27)

From the second inequality (29) it follows that the chemical compound number 16 (H = 1.174 bit) is incompatible with the other units of the subset. After excluding this element, we obtained the following statistics for the information function:

N₂ = 42,     = 1.87 ± 0.05,     = 1.457,    = 2.249,    S_H2 = 0.17.                                                       (28)

This subset is close to a normal distribution: . Criterion of Wilk-Shapiro exceeds the critical value:   W = 0.964 >   = 0.942. The examination of the subset uniformity leads to the following inequalities:

                                                                                             (29)

Thus, the subset comprises only compatible units.

Let's see whether the distinction between the average values of  and statistically significant. We predefine the distinction between the variances of  and :  = 1.52 < . That is, the distinction in variance is not statistically significant. Therefore, we must use the following inequality:

                                                                                                (30)                                                                                                                               

  N = 99,        N₁ =57,      N₂ = 42,     S_H1 = 0.21,    S_H2 = 0.17 .                     

The inequality (30) rejects the null hypothesis on equality of the average values of the information functions.

Again, we will use the association method of qualitative signs. We choose as the boundary value the following value of the information function (23):  = 1.80bit. The numerical data are contained in Table 3.

We can get additional information about the nonlinear dependence of H(Z) (Fig. 2) from a variational series of the grouped chemicals. It's typically used 6-8 groups for the sample size N ≈ 40-60. You must first make a ranging of the variational series (for example, in ascending of  Z). It is convenient to make groups at regular intervals. We chose the number of groups equal to n = 6.  Using the approximate relation  we can roughly determine the width of the interval group. Next, we find the group averages  and   for each classified data. Here i is the group number. Then we compare the ratio of the difference between the average values:

,             

                                                                                        (34)

,        .

The subscripts indicate the number of groups. Parameter  should be close to a constant value for the linear approximation. The frequency of the elements in groups (3₍₁₎, 9₍₂₎, 10₍₃₎, 13₍₄₎, 5₍₅₎, 3₍₆₎) is close to the normal distribution: W = 0.902 > = 0.788.

Separation of sample units into groups allows you to calculate the empirical correlation ratio = 0.84. Here  is the between-group variance;  is a total variance of the original sample of 43 units.  Obviously, the empirical relation  changes from zero to one and allows us to quantify the effect of Z factor on the variation of resulting character of H.

 Then we can calculate the theoretical correlation ratio . Here the value of = 0.02 is the variance of equalized values of the information function; S² = 0.025 is the variance of empirical (the facts) values of the information function. The theoretical correlation ratio is equal to = 0.89 (coefficient of determination is equal to = 0.79). That is, the change of the factor Z (79%) leads to the change of the information function. The remaining change in the information function H (21%) is explained by other factors that were not considered in the model. The nonlinear interrelation between the signs is strong if the inequality 0.7< < 0.9 (scale of Chaddock) is met.

Figure 3 shows the interrelationship of the information function and the electronic factor when changing the number of atomic groups CH₂ in molecules. RMSE values are so small, that the interrelation between factors Z and H come close to a functional interrelation.

As analysis has shown the information function relates to the value of π.  The value of π = 0.52 [14] defines an additional contribution of the group atoms CH₂ in hydrophobicity of molecules. Figure 4 shows this relationship for radioprotectors: CH₃(CH₂)_mNHCH₂CH₂SSO₃H (m = 0, 1, …, 17),

CH₃(CH₂)_mNH(CH₂)_nSPO₃H₂ (m = 2, 3, 4; n = 2, 3), NH₂(CH₂)_mSH (m = 2, 3, 4, 5).

The positive interrelation between the signs of Z and H is not random. Information function determines the diversity of the molecular structure, which in turn is determined by the number of different atoms, forming a bound complex of atoms, i.e., molecules. At the same time, the structure of the molecule is not an arbitrary set of various atoms, but is determined by the valence electrons in the outer electron shell. Apparently, this quantum-chemical property establishes the interrelation of two signs of Z and H for molecular structures.

Some distinctions between effective and inefficient radioprotectors we can get if we will analyze the frequency of the atoms appearnce in the molecule. Figure 5 shows the frequency of occurrence of atoms (C, H, N, O, S, P) in the molecule.

Using the data of Table 1, we can approximately indicate the frequency of occurrence of atoms in a molecule of hypothetical effective agent (for a homogeneous sample): P ~ 1, S ~ 1, N ~ 2, O ~ 3, C ~ 5-6,  H ~ 17 (Fig.5)^[1]. At the same time the most probable distribution of atoms in the inefficient agents (hypothetical molecule) will be as follows: P ~ 1, N ~ 1, O ~ 1, S ~ 2, C ~ 4, H ~ 8-10.

CONCLUSION

The proposed classification rules allow to identify the similarities between the molecular structures. These rules can be practically useful in a preliminary forecast of bioactivity of new chemical compounds. It should be noted that for the calculation of signs of Z and H is only required the knowledge of the chemical structural formula. This makes much easier the work in a preliminary searching for new bioactive chemicals. Classification rules allow you to set whether you can expect from a chemical compound effective biological action. The ability to separate the biologically active chemical compounds from the inactive chemical compounds on the basis of the sign of Z, apparently is due to the existence of the real molecular electrostatic potential. The magnitude of this potential varies from molecule to molecule. Moreover, there is a threshold of the electrostatic potential for effective chemical compounds which is lower of some value (in absolute value). The method described in this article, has yielded positive results when researching antifungal activity and toxicity of chemical compounds [15]. This method was also used in the analysis of the activity of carcinogenic chemicals [16].

However, it should be noted that these rules are not sensitive to iso-electronic molecular systems, as well as for the isomer molecules. This approach gives the most reliable results when analyzing the homologous series of chemical compounds. Homologous series are generally characterized by the signs that satisfy the compatibility condition.